Question

classify each number below as a rational number or an irrational number.
81.\overline{73} \quad \sqrt{64} \quad 15\pi \quad -74.83 \quad \sqrt{2}

Explanation:

Step1: Recall definitions

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. It includes terminating, repeating decimals, integers, fractions. An irrational number is a non - repeating, non - terminating decimal, like $\pi$, $\sqrt{2}$ (for non - perfect squares).

Step2: Classify $81.\overline{73}$

$81.\overline{73}$ is a repeating decimal (the 73 repeats). Repeating decimals can be expressed as a fraction. So it is rational.

Step3: Classify $\sqrt{64}$

$\sqrt{64}=8$, and 8 is an integer. Integers are rational (since $8 = \frac{8}{1}$). So $\sqrt{64}$ is rational.

Step4: Classify $15\pi$

$\pi$ is an irrational number (non - repeating, non - terminating). When we multiply an irrational number by a non - zero integer (15 here), the result is still irrational. So $15\pi$ is irrational.

Step5: Classify $- 74.83$

$-74.83$ is a terminating decimal. Terminating decimals can be written as a fraction (e.g., $-74.83=-\frac{7483}{100}$). So it is rational.

Step6: Classify $\sqrt{2}$

2 is not a perfect square. The square root of a non - perfect square is an irrational number (non - repeating, non - terminating decimal). So $\sqrt{2}$ is irrational.

Answer:

• $81.\overline{73}$: rational
• $\sqrt{64}$: rational
• $15\pi$: irrational
• $-74.83$: rational
• $\sqrt{2}$: irrational

To mark the table:

• For $81.\overline{73}$: mark the "rational" circle.
• For $\sqrt{64}$: mark the "rational" circle.
• For $15\pi$: mark the "irrational" circle.
• For $-74.83$: mark the "rational" circle.
• For $\sqrt{2}$: mark the "irrational" circle.